View source for AKNS-hierarchy

''Ablowitz–Kaup–Newell–Segur hierarchy''

An infinite tower of non-linear evolution equations that derives its name from the simplest non-trivial system of equations contained in it, the AKNS-equations

<table class="eq" style="width:100%;"> <tr><td valign="top" style="width:94%;text-align:center;"><img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/a/a130/a130130/a1301301.png" /></td> <td valign="top" style="width:5%;text-align:right;">(a1)</td></tr></table>

It were M.J. Ablowitz, D.J. Kaup, A.C. Newell, and H. Segur who showed that the initial value problem of this system of equations (cf. also [[Differential equation, partial, discontinuous initial (boundary) conditions|Differential equation, partial, discontinuous initial (boundary) conditions]]) could be solved with the inverse scattering transform (cf. also [[Korteweg–de Vries equation|Korteweg–de Vries equation]]). To get a natural embedding of the AKNS-equations in a larger system, one rewrites (a1) in zero-curvature form as

<table class="eq" style="width:100%;"> <tr><td valign="top" style="width:94%;text-align:center;"><img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/a/a130/a130130/a1301302.png" /></td> <td valign="top" style="width:5%;text-align:right;">(a2)</td></tr></table>

where

<table class="eq" style="width:100%;"> <tr><td valign="top" style="width:94%;text-align:center;"><img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/a/a130/a130130/a1301303.png" /></td> </tr></table>

Consider now the following polynomial expressions in the parameter <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/a/a130/a130130/a1301304.png" />:

<table class="eq" style="width:100%;"> <tr><td valign="top" style="width:94%;text-align:center;"><img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/a/a130/a130130/a1301305.png" /></td> <td valign="top" style="width:5%;text-align:right;">(a3)</td></tr></table>

<table class="eq" style="width:100%;"> <tr><td valign="top" style="width:94%;text-align:center;"><img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/a/a130/a130130/a1301306.png" /></td> </tr></table>

where the <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/a/a130/a130130/a1301307.png" /> are <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/a/a130/a130130/a1301308.png" />-valued functions depending on the variables <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/a/a130/a130130/a1301309.png" />, <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/a/a130/a130130/a13013010.png" />, and <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/a/a130/a130130/a13013011.png" /> and <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/a/a130/a130130/a13013012.png" />. For these data the zero-curvature equations read

<table class="eq" style="width:100%;"> <tr><td valign="top" style="width:94%;text-align:center;"><img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/a/a130/a130130/a13013013.png" /></td> <td valign="top" style="width:5%;text-align:right;">(a4)</td></tr></table>

<table class="eq" style="width:100%;"> <tr><td valign="top" style="width:94%;text-align:center;"><img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/a/a130/a130130/a13013014.png" /></td> </tr></table>

which is an infinite tower of equations extending the system (a1). The system (a4) generalizes from <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/a/a130/a130130/a13013015.png" /> to a general simple complex [[Lie algebra|Lie algebra]] <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/a/a130/a130130/a13013016.png" />, a [[Regular element|regular element]] <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/a/a130/a130130/a13013017.png" /> in a [[Cartan subalgebra|Cartan subalgebra]] <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/a/a130/a130130/a13013018.png" /> of <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/a/a130/a130130/a13013019.png" /> and an element <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/a/a130/a130130/a13013020.png" /> in <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/a/a130/a130130/a13013021.png" />, see [[#References|[a4]]] and [[#References|[a10]]]. Solutions of the equations (a4) can be obtained by the Zakharov–Shabat dressing method (cf. also [[Soliton|Soliton]]). Namely, consider the function

<table class="eq" style="width:100%;"> <tr><td valign="top" style="width:94%;text-align:center;"><img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/a/a130/a130130/a13013022.png" /></td> <td valign="top" style="width:5%;text-align:right;">(a5)</td></tr></table>

<table class="eq" style="width:100%;"> <tr><td valign="top" style="width:94%;text-align:center;"><img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/a/a130/a130130/a13013023.png" /></td> </tr></table>

with <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/a/a130/a130130/a13013024.png" /> belonging to the loop group <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/a/a130/a130130/a13013025.png" />. If this <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/a/a130/a130130/a13013026.png" /> factorizes as <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/a/a130/a130130/a13013027.png" />, with

<table class="eq" style="width:100%;"> <tr><td valign="top" style="width:94%;text-align:center;"><img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/a/a130/a130130/a13013028.png" /></td> <td valign="top" style="width:5%;text-align:right;">(a6)</td></tr></table>

<table class="eq" style="width:100%;"> <tr><td valign="top" style="width:94%;text-align:center;"><img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/a/a130/a130130/a13013029.png" /></td> </tr></table>

then conjugating the trivial connections (cf. also [[Connections on a manifold|Connections on a manifold]]) <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/a/a130/a130130/a13013030.png" /> and <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/a/a130/a130130/a13013031.png" /> with <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/a/a130/a130130/a13013032.png" /> gives connections of the required form:

<table class="eq" style="width:100%;"> <tr><td valign="top" style="width:94%;text-align:center;"><img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/a/a130/a130130/a13013033.png" /></td> <td valign="top" style="width:5%;text-align:right;">(a7)</td></tr></table>

<table class="eq" style="width:100%;"> <tr><td valign="top" style="width:94%;text-align:center;"><img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/a/a130/a130130/a13013034.png" /></td> </tr></table>

Since flatness is preserved by this procedure, this leads to solutions of (a4). If <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/a/a130/a130130/a13013035.png" />, as in the AKNS-case, one can take just as well <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/a/a130/a130130/a13013036.png" />.

It was observed by H. Flaschka, Newell and T. Ratiu [[#References|[a5]]] that the equations (a4) for the <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/a/a130/a130130/a13013037.png" />-case could be captured in the system

<table class="eq" style="width:100%;"> <tr><td valign="top" style="width:94%;text-align:center;"><img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/a/a130/a130130/a13013038.png" /></td> <td valign="top" style="width:5%;text-align:right;">(a8)</td></tr></table>

for the single series

<table class="eq" style="width:100%;"> <tr><td valign="top" style="width:94%;text-align:center;"><img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/a/a130/a130130/a13013039.png" /></td> <td valign="top" style="width:5%;text-align:right;">(a9)</td></tr></table>

<table class="eq" style="width:100%;"> <tr><td valign="top" style="width:94%;text-align:center;"><img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/a/a130/a130130/a13013040.png" /></td> </tr></table>

They showed that these equations are commuting Hamiltonian flows (cf. also [[Hamiltonian system|Hamiltonian system]]) on the Lie algebra <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/a/a130/a130130/a13013041.png" />, <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/a/a130/a130130/a13013042.png" />, with respect to natural [[Poisson brackets|Poisson brackets]]. Further, they introduced the flux tensor <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/a/a130/a130130/a13013043.png" /> by

<table class="eq" style="width:100%;"> <tr><td valign="top" style="width:94%;text-align:center;"><img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/a/a130/a130130/a13013044.png" /></td> <td valign="top" style="width:5%;text-align:right;">(a10)</td></tr></table>

<table class="eq" style="width:100%;"> <tr><td valign="top" style="width:94%;text-align:center;"><img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/a/a130/a130130/a13013045.png" /></td> </tr></table>

and proved the local conservation laws of the system, namely

<table class="eq" style="width:100%;"> <tr><td valign="top" style="width:94%;text-align:center;"><img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/a/a130/a130130/a13013046.png" /></td> <td valign="top" style="width:5%;text-align:right;">(a11)</td></tr></table>

The left-hand side of (a11) is, in fact, even symmetric under permutations of the indices <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/a/a130/a130130/a13013047.png" />, <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/a/a130/a130130/a13013048.png" />, <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/a/a130/a130130/a13013049.png" /> and this property made them introduce a potential <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/a/a130/a130130/a13013050.png" /> by

<table class="eq" style="width:100%;"> <tr><td valign="top" style="width:94%;text-align:center;"><img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/a/a130/a130130/a13013051.png" /></td> <td valign="top" style="width:5%;text-align:right;">(a12)</td></tr></table>

The equations (a8) are called the Lax equations of the AKNS-hierarchy. As such, the AKNS-hierarchy is a natural reduction of the two-component KP-hierarchy (cf. also [[KP-equation|KP-equation]]; [[#References|[a7]]]), a fact that enables a description in the Grassmannian of that hierarchy.

It was shown by M.J. Bergvelt and A.P.E. ten Kroode [[#References|[a1]]] that it is natural to consider a system of zero-curvature relations (a4) labelled by the root lattice of the Lie algebra, where the operators at different sites of the lattice are linked by Toda-type differential-difference equations. For example, for nearest neighbour sites there holds

<table class="eq" style="width:100%;"> <tr><td valign="top" style="width:94%;text-align:center;"><img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/a/a130/a130130/a13013052.png" /></td> <td valign="top" style="width:5%;text-align:right;">(a13)</td></tr></table>

where

<table class="eq" style="width:100%;"> <tr><td valign="top" style="width:94%;text-align:center;"><img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/a/a130/a130130/a13013053.png" /></td> </tr></table>

This phenomenon is due to the fact that there is a natural lattice group that commutes with the commuting flows corresponding to the parameters <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/a/a130/a130130/a13013054.png" />.

In the representation-theoretic approach to soliton equations (see [[#References|[a3]]], [[#References|[a8]]]), the soliton equations occur as the equations describing the group orbit of the highest weight vector. A similar description holds for these combined differential-difference equations. Let <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/a/a130/a130130/a13013055.png" /> be the basic representation of the Kac–Moody Lie algebra <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/a/a130/a130130/a13013056.png" /> (cf. also [[Kac–Moody algebra|Kac–Moody algebra]]). In <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/a/a130/a130130/a13013057.png" /> one takes the homogeneous Heisenberg algebra

<table class="eq" style="width:100%;"> <tr><td valign="top" style="width:94%;text-align:center;"><img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/a/a130/a130130/a13013058.png" /></td> </tr></table>

where <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/a/a130/a130130/a13013059.png" /> is the central element of <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/a/a130/a130130/a13013060.png" /> that is in the kernel of the projection of <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/a/a130/a130130/a13013061.png" /> onto the loop algebra of <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/a/a130/a130130/a13013062.png" />. The <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/a/a130/a130130/a13013063.png" />-module <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/a/a130/a130130/a13013064.png" /> decomposes with respect to the action of the homogeneous Heisenberg algebra <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/a/a130/a130130/a13013065.png" /> as a direct sum of irreducible <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/a/a130/a130130/a13013066.png" />-modules <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/a/a130/a130130/a13013067.png" /> labelled by the root lattice. Thus, one can write each element of <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/a/a130/a130130/a13013068.png" /> as <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/a/a130/a130130/a13013069.png" />. The group orbit of the highest weight vector can then be characterized by a set of so-called Hirota bilinear relations for the components <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/a/a130/a130130/a13013070.png" />. By using the representation theory one constructs a series of elements <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/a/a130/a130130/a13013071.png" /> in a suitable completion of the Kac–Moody group associated with <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/a/a130/a130130/a13013072.png" /> such that the vacuum expectation value of <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/a/a130/a130130/a13013073.png" /> is exactly <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/a/a130/a130130/a13013074.png" />. The Birkhoff decomposition of the <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/a/a130/a130130/a13013075.png" /> in that group then enables one to construct solutions of the lattice zero-curvature equations, [[#References|[a2]]]. In particular, the operators <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/a/a130/a130130/a13013076.png" /> from (a13) obtained in this way can be expressed in the components <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/a/a130/a130130/a13013077.png" /> by

<table class="eq" style="width:100%;"> <tr><td valign="top" style="width:94%;text-align:center;"><img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/a/a130/a130130/a13013078.png" /></td> <td valign="top" style="width:5%;text-align:right;">(a14)</td></tr></table>

By using the adjoint action of the Kac–Moody group, Bergvelt and ten Kroode also showed [[#References|[a2]]] that the

<table class="eq" style="width:100%;"> <tr><td valign="top" style="width:94%;text-align:center;"><img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/a/a130/a130130/a13013079.png" /></td> <td valign="top" style="width:5%;text-align:right;">(a15)</td></tr></table>

give exactly the flux tensor from (a10), thus furnishing a representation-theoretic basis for the results in [[#References|[a5]]].

A geometric way to look at <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/a/a130/a130130/a13013081.png" />-functions, see e.g. [[#References|[a9]]], is to consider a [[Homogeneous space|homogeneous space]] over the relevant loop group <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/a/a130/a130130/a13013082.png" />, a holomorphic line [[Bundle|bundle]] <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/a/a130/a130130/a13013083.png" /> over this space and its pull-back over the corresponding central extension <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/a/a130/a130130/a13013084.png" /> of <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/a/a130/a130130/a13013085.png" />. If this last line bundle has a global holomorphic section, then <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/a/a130/a130130/a13013086.png" />-functions measure the failure of equivariance of partial liftings from <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/a/a130/a130130/a13013087.png" /> to <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/a/a130/a130130/a13013088.png" /> with respect to this section. With this point of view, one can also arrive at the formulas in (a14) by lifting the discrete group of transformations that commute with the flows from (a6) appropriately, see [[#References|[a11]]].

An important class of equations associated with the AKNS-hierarchy are the so-called stationary AKNS-equations. These are the differential equations for the functions <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/a/a130/a130130/a13013089.png" /> and <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/a/a130/a130130/a13013090.png" /> from the first-order differential operator

<table class="eq" style="width:100%;"> <tr><td valign="top" style="width:94%;text-align:center;"><img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/a/a130/a130130/a13013091.png" /></td> </tr></table>

resulting from the existence of a <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/a/a130/a130130/a13013092.png" />-matrix-valued differential operator

<table class="eq" style="width:100%;"> <tr><td valign="top" style="width:94%;text-align:center;"><img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/a/a130/a130130/a13013093.png" /></td> </tr></table>

that commutes with <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/a/a130/a130130/a13013094.png" />. Such a pair is naturally associated with a [[Hyper-elliptic curve|hyper-elliptic curve]] of genus <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/a/a130/a130130/a13013095.png" /> and that is why one calls <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/a/a130/a130130/a13013096.png" /> an algebro-geometric AKNS-potential. The elliptic algebro-geometric AKNS-potentials have been characterized in [[#References|[a6]]]. They correspond exactly to the potentials for which the equation <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/a/a130/a130130/a13013097.png" /> has a meromorphic [[Fundamental system of solutions|fundamental system of solutions]] with respect to <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/a/a130/a130130/a13013098.png" /> for all values of the spectral parameter <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/a/a130/a130130/a13013099.png" />.

====References====
<table><TR><TD valign="top">[a1]</TD> <TD valign="top">  M.J. Bergvelt,   A.P.E. ten Kroode,   "Differential-difference <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/a/a130/a130130/a130130100.png" /> equations and homogeneous Heisenberg algebras"  ''J. Math. Phys.'' , '''28'''  (1987)  pp. 302–306</TD></TR><TR><TD valign="top">[a2]</TD> <TD valign="top">  M.J. Bergvelt,   A.P.E. ten Kroode,   "<img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/a/a130/a130130/a130130101.png" />-functions and zero curvature equations of Toda–AKNS type"  ''J. Math. Phys.'' , '''29'''  (1988)  pp. 1308–1320</TD></TR><TR><TD valign="top">[a3]</TD> <TD valign="top">  E. Date,   M. Jimbo,   M. Kashiwara,   T. Miwa,   "Transformation groups for soliton equations" , ''Non-linear Integrable Systems; Classical Theory and Quantum Theory (Proc. RIMS Symp.)'' , World Sci.  (1983)  pp. 41–119</TD></TR><TR><TD valign="top">[a4]</TD> <TD valign="top">  V.G. Drinfel'd,   V.V. Sokolov,   "Lie algebras and equations of Korteweg de Vries type"  ''Itogi Nauki i Tekhn. Ser. Sovrem. Probl. Mat.'' , '''24'''  (1984)  pp. 81–180  (In Russian)</TD></TR><TR><TD valign="top">[a5]</TD> <TD valign="top">  H. Flaschka,   A.C. Newell,   T. Ratiu,   "Kac–Moody Lie algebras and soliton equations II. Lax equations associated with <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/a/a130/a130130/a130130102.png" />"  ''Physica'' , '''9D'''  (1983)  pp. 300–323</TD></TR><TR><TD valign="top">[a6]</TD> <TD valign="top">  F. Gesztesy,   R. Weikard,   "A characterization of all elliptic algebro-geometric solutions of the AKNS hierarchy"  ''Acta Math.'' , '''181'''  (1998)  pp. 63–108</TD></TR><TR><TD valign="top">[a7]</TD> <TD valign="top">  G.F. Helminck,   G.F. Post,   "A convergent framework for the multicomponent <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/a/a130/a130130/a130130103.png" />-hierarchy"  ''Trans. Amer. Math. Soc.'' , '''324''' :  1  (1991)  pp. 271–292</TD></TR><TR><TD valign="top">[a8]</TD> <TD valign="top">  V.G. Kac,   "Infinite dimensional Lie algebras" , Cambridge Univ. Press  (1989)  (Edition: Third)</TD></TR><TR><TD valign="top">[a9]</TD> <TD valign="top">  G. Segal,   G. Wilson,   "Loop groups and equations of KdV type"  ''Publ. Math. IHES'' , '''63'''  (1985)  pp. 1–64</TD></TR><TR><TD valign="top">[a10]</TD> <TD valign="top">  G. Wilson,   "The modified Lax and two-dimensional Toda lattice equations associated with simple Lie algebras"  ''Ergod. Th. Dynam. Syst.'' , '''1'''  (1981)  pp. 361–380</TD></TR><TR><TD valign="top">[a11]</TD> <TD valign="top">  G. Wilson,   "The <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/a/a130/a130130/a130130104.png" />-functions of the AKNS equations" , ''Integrable Systems: the Verdier Memorial Conf. (Actes Colloq. Internat. Luminy)'' , ''Progress in Math.'' , '''115'''  (1993)  pp. 131–145</TD></TR></table>